Stable hypersurfaces as minima of the integral of an anisotropic mean curvature preserving a linear combination of area and volume

作者：Gervasio Colares A; Floriano da Silva J

来源：Mathematische Zeitschrift, 2013, 275(1-2): 595-623.

DOI：10.1007/s00209-013-1149-6

• Hypersurfaces
• Constant anisotropic mean curvature
• Variational problem
• Stability
• Wulff shape

We deal with a compact hypersurface without boundary immersed in Euclidean space with the quotient of anisotropic mean curvatures , for real numbers and . Such a hypersurface is a critical point for the variational problem preserving a linear combination (with coefficientes and ) of the -area and the -volume enclosed by . We show that is -stable if and only if, up to translations and homotheties, it is the Wulff shape of , under some assumptions on and proved to be sharp. For and , this gives the known -stability of the -area for volume preserving variations; if also it yields the stability studied by Alencar-do Carmo-Rosenberg and Barbosa-Colares. For we also prove a characterization of the Wulff shape as a critical point of the -area for variations preserving the -area, , without the -stability hypothesis.

• 出版日期2013-10